threeTwoCosPath
plain-language theorem explainer
Complex split-form dihedral cosine of the hinge opposite vertex pair (p,q) along the physical Wick arc of the (3,2) causal 4-simplex (a=α=1). Gravity/CDT workers cite it as the model path for all ten threeTwo hinge certificates. One-line composition of the complex edge continuation with the cofactor cosine.
Claim. For opposite vertices $p,q\in\{0,\ldots,4\}$ and arc parameter $t\in\mathbb{R}$, let $x(t)$ be the complex squared-edge 10-tuple of the $(3,2)$ causal 4-simplex along the physical Wick arc ($a=1$, $\alpha=1$). Then $\mathrm{cos}_{p,q}(t)$ is the split-form cofactor dihedral cosine of the triangular hinge opposite $\{p,q\}$ evaluated at $x(t)$.
background
Lane B2 of the QG Seven-Gaps campaign treats all-hinge complex-first Wick continuation of the (3,2) causal 4-simplex. In 4d CDT there are two simplex types between adjacent slices; threeTwo places three vertices on slice $t$ and two on $t+1$, so the six cross edges are timelike and the rest spacelike.
The complex edge path keeps spacelike squared lengths at $a^2$ and sends timelike edges along the canonical upper-half-plane arc. The split-form dihedral cosine at opposite pair $(p,q)$ is the ratio of the Cayley-Menger cofactor $C_{pq}$ to the split denominator built from the two face cofactors; the $+C_{pq}$ convention matches the 3D regular-tetrahedron value $+1/3$ and the 4D regular-simplex interior cosine $-1/4$.
Hinges fall into three classes by opposite pair: the single spacelike hinge $(0,1,2)$, six mixed lower-upper pairs, and three upper-pair hinges inside the lower triple. Closed cofactor forms for each class are kernel-checked by $5\times 5$ minors in this module.
proof idea
Pure definitional wrapper. Instantiate the complex squared-edge continuation at type threeTwo with physical parameters $a=1$, $\alpha=1$ and arc time $t$, then feed that 10-tuple into the split-form cofactor cosine at opposite indices $(p,q)$. No algebraic simplification or proof obligation beyond the two upstream defs.
why it matters
This is the model path every threeTwo hinge certificate evaluates. Downstream boundary theorems (boundary32_pair01 through the full ten-pair table) prove continuous extension on $[0,1]$ with explicit Lorentzian and Euclidean endpoint values, each by specializing this path. The B3 headline wick_hinge_data_continuation_complete quantifies over both simplex types and every opposite pair, requiring branch-regularity on the open arc and continuous paths ending at the Euclidean regular-4-simplex cosine $-1/4$; the threeTwo half of that statement is built on this def.
It sits inside the finishing charter for complex-first Wick continuation of causal 4-simplex hinges, the geometric substrate for the Recognition gravity lane. The spacelike hinge's Lorentzian endpoint lands on the arccos cut (classical boost angle), so branch certificates are interior-only while the split rational form remains exact at the endpoint.
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